First of, I'm new to signal processing. I have a signal which is a linear composition of several basis signals, whereas the same basis signal can occur several times, that is translated, but not scaled. These basis signals look very similar to wavelets. I'm thinking of using a continuous wavelet transform to get the decomposition of the signal with respect the basis signals. However, I assume that a set of basis signals needs to fulfill certain properties to be considered a wavelet family. Could you please point out to me what these may be?
Your question is quite central to the development of wavelet theory.
Indeed, the word wavelet has an early history. It is coined in "The form and nature of seismic waves and the structure of seismograms", Norman Ricker, 1940, Geophysics. Here, the wavelet is quite close to your problem, and the scale factor is not an issue. Later, following Jean Morlet, wavelets have become scale-related transformations.
Resultingly, following @Batman advices, I would relate your question to matched filtering, basis pursuit or deconvolution/source separation.
Yet, you can perform matched filtering of basis signal templates in a wavelet domain, which can help in the case of noise, and may simplify the local estimation of adapted matched filters. An example in seismic signal processing is given in Adaptive multiple subtraction with wavelet-based complex unary Wiener filters, 2012 with continuous complex wavelet frames, or in a more complete optimization paradigm with bounds on sparsity in A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal, 2014.
What you should made clearer is you really want to do with your signal.
It sounds like you don't need a wavelet. A wavelet's translation and dilation properties allow you to get both time and frequency resolution.
A matched filter takes an inner product of a signal with another signal, allowing you to detect a known signal in noise (by correlating the signal you have with the known signal). You may want to use this idea instead.
Another idea is matching pursuit type algorithms -- if you have an overcomplete set of signals, and you want to find an approximation of a signal with respect to these overcomplete set of signals, you can do this through a matching pursuit algorithm. Basically, iteratively form a residual by finding the largest inner product signal in the set and then subtracting a scaled verison of the signal.